Figure 1.16 shows a single robot joint link driven through a gear ratio armature-controlled servo motor. The input is the armature voltage Va(t) and the output is the load-shaft angle O(t). Derive the mathematical model for this system: develop the circuit differential equation, the electromechanical coupling equations, and the rotational mechanical differential equation. Eliminate intermediate variables and simplify. It will be convenient to use the transfer-function approach. Assume the mass-moment of inertia of all outboard links plus any load JL6) is constant (a reasonable assumption when the gear ratio @Mfo1 is much larger than I2I, the cc of industrial robots). The parameters in Figure 1.16 are summarized below.
Derive a valid state-space description for the system in Figure 1.16. That is, specify the state variables and derive the coefficient matrices A, B, and C. Write out your results in matrix-vector form. Give the system order and matrix-vector dimensions of your result. Consider two distinct cases: Single-input, single-output: armature voltage VA(t) as the input and robot load-shaft angle O(t) as the output. Single-input, single-output: armature voltage VA(t) as the input and robot load-shaft angular velocity ML(t) as the output.